Non-parametric Algorithmic Generation of Neuronal Morphologies

Abstract

Generation algorithms allow for the generation of Virtual Neurons (VNs) from a small set of morphological properties. The set describes the morphological properties of real neurons in terms of statistical descriptors such as the number of branches and segment lengths (among others). The majority of reconstruction algorithms use the observed properties to estimate the parameters of a priori fixed probability distributions in order to construct statistical descriptors that fit well with the observed data. In this article, we present a non-parametric generation algorithm based on kernel density estimators (KDEs). The new algorithm is called KDE-Neuron and has three advantages over parametric reconstruction algorithms: (1) no a priori specifications about the distributions underlying the real data, (2) peculiarities in the biological data will be reflected in the VNs, and (3) ability to reconstruct different cell types. We experimentally generated motor neurons and granule cells, and statistically validated the obtained results. Moreover, we assessed the quality of the prototype data set and observed that our generated neurons are as good as the prototype data in terms of the used statistical descriptors. The opportunities and limitations of data-driven algorithmic reconstruction of neurons are discussed.

Appendix: Kernel Density Estimates

In the appendix we give more details on the implementation of kernel density estimation for univariate and bivariate prototype data. We also explain how the automatic selection of the bandwidth parameter has been performed which is an important issue that we did not address so far.

Univariate KDEs

An intuitive, non-parametric way to model observed data is by means of a histogram estimator, which is formally defined by:

$$ \hat{f}(x) = \frac{| \left\{ x_i \in N(x), \: i=1, \dots, n \right\} |}{nh} \enspace , $$

(5)

where N(x) is the bin centred at x and of width h. A KDE is a smooth version of the histogram estimator were a kernel function K is used that counts observations close to x with weights that are decreasing with the distance from x (Parzen 1962):

$$ \hat{f}(x) = \frac{1}{nh} \sum_{i=1}^n K\left(\frac{x - x_i}{h}\right) \enspace . $$

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A popular choice of kernel function is the Gaussian distribution with zero mean and unit variance. The KDE with Gaussian kernel function becomes:

$$ \hat{f}(x) = \frac{1}{nh\sqrt{2\pi}} \sum_{i=1}^n \exp\left[ -\frac{(x - x_i)^2}{2h^2} \right] \enspace , $$

(7)

with h the bandwidth (standard deviation) of the Gaussian components. The kernel density estimator is a legitimate probability density function when the kernel function is non-negative everywhere and integrates to one. It is easily verified that this holds for the Gaussian. The parameter h acts as the smoothing parameter.

Multivariate KDEs

Suppose we have a sample consisting of n bivariate observations of the form (x ₁, y ₁), ..., (x _n , y _n ) that are independently drawn from an unknown probability distribution f over the random variable (X, Y). The KDE for the joint distribution is then:

$$ \hat{f}(x, y) = \frac{1}{nh_xh_y} \sum_{i=1}^n K\left(\frac{x - x_i}{h_x}\right) K\left(\frac{y - y_i}{h_y}\right) \enspace , $$

(8)

with h _x and h _y the bandwidths for the random variables X and Y, respectively. Since we are using the Gaussian as kernel function the estimator for the joint distribution becomes:

$$ \hat{f}(x, y) = \frac{1}{nh_xh_y\sqrt{(2\pi)^2}} \sum_{i=1}^n \exp\left[ -\frac{(x - x_i)^2}{2h_x^2} -\frac{(y - y_i)^2}{2h_y^2} \right]. $$

(9)

In the rest of this appendix we write equations explicitly for the Gaussian kernel although they can be written in terms of any other kernel function.

The kernel density estimate of a marginal distribution can be calculated from an application of the sum rule of probability. For example, the estimator for the marginal distribution f _X (x) of X (assuming Y to be a real-valued random variable) is:

$$ \hat{f}_X(x) = \int^{+\infty}_{-\infty} \hat{f}(x, y) \text{d} y = \frac{1}{nh_x\sqrt{2\pi}} \sum_{i=1}^n \exp\left[ -\frac{(x - x_i)^2}{2h_x^2} \right], $$

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since the Gaussian integrates to one over the real line. The kernel density estimator for f _Y (y) can be defined analogously. A kernel density estimator for a conditional density, e.g., the estimator of the density function of X given that Y is some value y, is computed by a straightforward application of the product rule of probability:

$$ \hat{f}(x | y) = \frac{\hat{f}(x, y)}{\hat{f}_Y(y)} = \frac{ \frac{1}{h_x \sqrt{2\pi}} \sum_{i=1}^n \exp\left[ -\frac{(x - x_i)^2}{2h_x^2} -\frac{(y - y_i)^2}{2h_y^2} \right] } {\sum_{i=1}^n \exp\left[ -\frac{(y - y_i)^2}{2h_y^2} \right]}. $$

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The above equations for density estimation with respect to bivariate observations can be generalized in a straightforward manner to observations with more than two dimensions.

Automatic Bandwidth Selection

The bandwidth is the single parameter for kernel density estimation using Gaussian kernel. It acts as a smoothing factor and consequently determines the quality of the estimator. The best bandwidth has to be learned from the data since the data, besides prior knowledge if available, are the only objective basis to decide when the estimator is oversmoothing or undersmoothing. Various methods have been proposed for the task of automatic bandwidth selection. A brief survey is found in (Wand and Jones 1995; Jones et al. 1996).

It is natural to define the best bandwidth as the one resulting in the best KDE. The quality of a KDE $\hat{f}(x)$ with respect to the true density f(x) is often measured by the integrated squared error:

$$ \text{ISE}(\hat{f}, f{\kern1pt}) = \int^{+\infty}_{-\infty} (\hat{f}(x) - f(x))^2 dx \enspace . $$

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The ISE depends on a particular sample of n observations since the sample is used as information for the KDE. The mean integrated squared error averages the integrated squared error over all samples:

$$ \text{MISE}(\hat{f}, f{\kern1pt}) = \int^{+\infty}_{-\infty} \text{E}\left[(\hat{f}(x) - f(x))^2\right] dx \enspace , $$

(13)

where we have used $\text{E}\left[\cdot\right]$ to denote expected value. Clearly, the bandwidth with minimum MISE gives us an estimator that on average has lowest ISE for a particular sample of n observations. Given that n is sufficiently large, an approximation to MISE can be shown to be (Scott 1992):

$$ \text{AMISE}(\hat{f}, f) = \frac{R(K)}{nh} + \frac{h^4\mu_2(K)^2R(f'')}{4} \enspace , $$

(14)

where K is the kernel and the following definitions are used:

$$ R(g) = \int^{+\infty}_{-\infty} g(x)^2 dx \mbox{, and } \mu_2(g) = \int^{+\infty}_{-\infty} x^2g(x) dx \enspace . $$

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Setting the derivative of Eq. 14 with respect to the bandwidth to zero gives the optimal solution for the bandwidth:

$$ h = \left[ \frac{R(K)}{\mu_2(K)^2 R(f'') n} \right]^{1/5} \enspace . $$

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Unfortunately this solution depends on the second order derivative of the true density f(x) which is unknown. Moreover, estimating the derivative is a more difficult problem than estimating the density itself (Raykar and Duraiswami 2006).

There are several approaches to approximate the best bandwidth in terms of AMISE. The basic approach is called rules of thumb and assumes that the data is generated by a Gaussian distribution (Wand and Jones 1995). Given data of dimensionality d, the optimal bandwidth for the i-th variable is then given by:

$$ h_i = \left( \frac{4}{d+2} \right)^{1/(d+4)} n^{-1/(d+4)} \hat{\sigma}_i \enspace , $$

(17)

with $\hat{\sigma}_i$ the estimator for the standard deviation of the i-th variable. For univariate data (d = 1), a more specific estimate of the bandwidth is:

$$ h_i = 0.9 \: S \: n^{-1/5} \enspace , $$

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with S the minimum of the standard deviation and three-quarters of its interquartile range (Silverman 1986). In our experiments we found that the rules of thumb bandwidths for multivariate data are often better than the bandwidths found by some more advanced approaches, even though the assumption of Gaussian distributions is not satisfied. This is in line with the findings reported by Loader (1999).